blob: f8139bf92041c274ead23211bfd22b0965e68de3 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
|
Require Export Bedrock.Word Bedrock.Nomega.
Require Import NPeano NArith PArith Ndigits Compare_dec Arith.
Require Import ProofIrrelevance Ring List.
Require Import MultiBoundedWord.
Section Vector.
Import ListNotations.
Definition vec T n := {x: list T | length x = n}.
Definition vget {n T} (x: vec T n) (i: {v: nat | (v < n)%nat}): T.
refine (
match (proj1_sig x) as x' return (proj1_sig x) = x' -> _ with
| [] => fun _ => _
| x0 :: xs => fun _ => nth (proj1_sig i) (proj1_sig x) x0
end eq_refl); abstract (
destruct x as [x xp], i as [i ip]; destruct x as [|x0 xs];
simpl in xp; subst; inversion _H; clear _H; omega).
Defined.
Lemma vget_spec: forall {T n} (x: vec T n) (i: {v: nat | (v < n)%nat}) (d: T),
vget x i = nth (proj1_sig i) (proj1_sig x) d.
Proof.
intros; destruct x as [x xp], i as [i ip];
destruct x as [|x0 xs]; induction i; unfold vget; simpl;
intuition; try (simpl in xp; subst; omega);
induction n; simpl in xp; try omega; clear IHi IHn.
apply nth_indep.
assert (length xs = n) by omega; subst.
omega.
Qed.
Definition vec0 {T} : vec T 0.
refine (exist _ [] _); abstract intuition.
Defined.
Lemma lift0: forall {T n} (x: T), vec (word n) 0 -> T.
intros; refine x.
Defined.
Lemma liftS: forall {T n m} (f: vec (word n) m -> word n -> T),
(vec (word n) (S m) -> T).
Proof.
intros T n m f v; destruct v as [v p]; induction m, v;
try (abstract (inversion p)).
- refine (f (exist _ [] _) w); abstract intuition.
- refine (f (exist _ v _) w); abstract intuition.
Defined.
Lemma vecToList: forall T n m (f: vec (word n) m -> T),
(list (word n) -> option T).
Proof.
intros T n m f x; destruct (Nat.eq_dec (length x) m).
- refine (Some (f (exist _ x _))); abstract intuition.
- refine None.
Defined.
End Vector.
Ltac vectorize :=
repeat match goal with
| [ |- context[?a - 1] ] =>
let c := eval simpl in (a - 1) in
replace (a - 1) with c by omega
| [ |- vec (word ?n) O -> ?T] => apply (@lift0 T n)
| [ |- vec (word ?n) ?m -> ?T] => apply (@liftS T n (m - 1))
end.
Section Examples.
Lemma vectorize_example: (vec (word 32) 2 -> word 32).
vectorize; refine (@wplus 32).
Qed.
End Examples.
|
